Finite Group Databasebeta

← C22×C61×#557 #29566142

Subgroups — C22×C61×#557 #29566142

Order 15616.

7414 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#557C2148yesmaximal, direct product factor
2C22×C4⋊Q8×C61C213yesmaximal
3C22×#125×C61C216yesmaximal
4C22×C4×Q8×C61C213yesmaximal
5C22×C4×Q8×C61C212yesmaximal
6C22×C4×Q8×C61C211yesmaximal, characteristic
7C23×Q8×C61C2213yes—
8C23×Q8×C61C2211yescharacteristic
9C61×#557C221256yesdirect product factor
10C2×C4⋊Q8×C61C22172yes—
11C2×#125×C61C221144yes—
12C2×C4×Q8×C61C22124yes—
13C2×C4×Q8×C61C22172yes—
14C2×C4×Q8×C61C22148yes—
15C22×C4⋊C4×C61C2211yescharacteristic
16C22×C4⋊C4×C61C2216yes—
17C22×C4⋊C4×C61C2213yes—
18C22×C4⋊C4×C61C2216yes—
19C22×C4⋊C4×C61C2216yes—
20C22×C42×C61C2216yes—
21C22×C42×C61C2213yes—
22C22×Q8×C61—224no—
23C22×Q8×C61Q8116yescomplemented
24C22×Q8×C61C23136yes—
25C22×Q8×C61C23112yes—
26C23×C4×C61C2316yes—
27C23×C4×C61C2312yes—
28C23×C4×C61C2313yes—
29C23×C4×C61C2313yes—
30C23×C4×C61C2311yescharacteristic
31C4⋊Q8×C61C231192yes—
32#125×C61C231384yes—
33C4×Q8×C61C231128yes—
34C4×Q8×C61C231192yes—
35C4×Q8×C61C23164yes—
36C2×C4⋊C4×C61C23172yes—
37C2×C4⋊C4×C61C23172yes—
38C2×C4⋊C4×C61C23112yes—
39C2×C4⋊C4×C61C23172yes—
40C2×C4⋊C4×C61C23136yes—
41C2×C42×C61C23136yes—
42C2×C42×C61C23172yes—
43C24×C61C2411yescharacteristic
44C2×Q8×C61—2144no—
45C2×Q8×C61C2×Q8196yescomplemented
46C2×Q8×C61C24148yes—
47C2×Q8×C61C24116yes—
48C22×C4×C61—224no—
49C22×C4×C61—28no—
50C22×C4×C61—212no—
51C22×C4×C61C2×Q8124yes—
52C22×C4×C61#4118yes—
53C22×C4×C61C24112yes—
54C22×C4×C61C24118yes—
55C22×C4×C61C24136yes—
56C22×C4×C61C24118yes—
57C22×C4×C61C2416yes—
58C4⋊C4×C61C24116yes—
59C4⋊C4×C61C24196yes—
60C4⋊C4×C61C24148yes—
61C4⋊C4×C61C24196yes—
62C4⋊C4×C61C24196yes—
63C42×C61C24196yes—
64C42×C61C24148yes—
65C23×C61C22×Q814yes—
66C23×C61C2×#4114yes—
67C23×C612−1+414yes—
68C23×C61C2513yes—
69Q8×C61—296no—
70Q8×C61C22×Q8164yescomplemented
71C2×C4×C61—272no—
72C2×C4×C61—224no—
73C2×C4×C61—236no—
74C2×C4×C61C22×Q8172yes—
75C2×C4×C61C2×#41124yes—
76C2×C4×C61C2518yes—
77C2×C4×C61C25112yes—
78C2×C4×C61C25124yes—
79C2×C4×C61C25112yes—
80C2×C4×C61C2514yes—
81C22×#557C6111yesmaximal, characteristic, direct product factor, Sylow
82C22×C61#557116yesdirect product factor
83C22×C61C23×Q816yes—
84C22×C61C22×#4116yes—
85C22×C61C2×2−1+416yes—
86C22×C61C2611yescharacteristic
87C4×C61—224no—
88C4×C61—28no—
89C4×C61—212no—
90C4×C61C23×Q8124yes—
91C4×C61C22×#4118yes—
92C2×#557C2×C61148yesdirect product factor
93C22×C4⋊Q8C2×C6113yes—
94C22×#125C2×C6116yes—
95C22×C4×Q8C2×C6113yes—
96C22×C4×Q8C2×C6111yescharacteristic
97C22×C4×Q8C2×C6112yes—
98C2×C61C2×#557112yesdirect product factor
99C2×C61C24×Q811yescharacteristic
100C2×C61C23×#4111yescharacteristic
101C2×C61C22×2−1+411yescharacteristic
102C23×Q8C22×C6113yes—
103C23×Q8C22×C6111yescharacteristic
104#557C22×C611256yesdirect product factor
105C2×C4⋊Q8C22×C61172yes—
106C2×#125C22×C611144yes—
107C2×C4×Q8C22×C61148yes—
108C2×C4×Q8C22×C61172yes—
109C2×C4×Q8C22×C61124yes—
110C22×C4⋊C4C22×C6111yescharacteristic
111C22×C4⋊C4C22×C6113yes—
112C22×C4⋊C4C22×C6116yes—
113C22×C4⋊C4C22×C6116yes—
114C22×C4⋊C4C22×C6116yes—
115C22×C42C22×C6116yes—
116C22×C42C22×C6113yes—
117C61C22×#55711yescharacteristic, direct product factor, Sylow
118C22×Q8—224no—
119C22×Q8Q8×C61116yescomplemented
120C22×Q8C23×C61136yes—
121C22×Q8C23×C61112yes—
122C23×C4C23×C6113yes—
123C23×C4C23×C6116yes—
124C23×C4C23×C6112yes—
125C23×C4C23×C6113yes—
126C23×C4C23×C6111yescharacteristic
127C4⋊Q8C23×C611192yes—
128#125C23×C611384yes—
129C4×Q8C23×C611128yes—
130C4×Q8C23×C611192yes—
131C4×Q8C23×C61164yes—
132C2×C4⋊C4C23×C61112yes—
133C2×C4⋊C4C23×C61172yes—
134C2×C4⋊C4C23×C61136yes—
135C2×C4⋊C4C23×C61172yes—
136C2×C4⋊C4C23×C61172yes—
137C2×C42C23×C61172yes—
138C2×C42C23×C61136yes—
139C24C24×C6111yescharacteristic
140C2×Q8—2144no—
141C2×Q8C2×Q8×C61196yescomplemented
142C2×Q8C24×C61148yes—
143C2×Q8C24×C61116yes—
144C22×C4—224no—
145C22×C4—212no—
146C22×C4—28no—
147C22×C4C2×Q8×C61124yes—
148C22×C4#41×C6118yes—
149C22×C4C24×C61112yes—
150C22×C4C24×C61118yes—
151C22×C4C24×C61136yes—
152C22×C4C24×C61118yes—
153C22×C4C24×C6116yes—
154C4⋊C4C24×C61116yes—
155C4⋊C4C24×C61196yes—
156C4⋊C4C24×C61148yes—
157C4⋊C4C24×C61196yes—
158C4⋊C4C24×C61196yes—
159C42C24×C61196yes—
160C42C24×C61148yes—
161C23C22×Q8×C6114yes—
162C23C2×#41×C6114yes—
163C232−1+4×C6114yes—
164C23C25×C6113yes—
165Q8—296no—
166Q8C22×Q8×C61164yescomplemented
167C2×C4—272no—
168C2×C4—224no—
169C2×C4—236no—
170C2×C4C22×Q8×C61172yes—
171C2×C4C2×#41×C61124yes—
172C2×C4C25×C6118yes—
173C2×C4C25×C61112yes—
174C2×C4C25×C61124yes—
175C2×C4C25×C61112yes—
176C2×C4C25×C6114yes—
177C22C61×#557116yesdirect product factor
178C22C23×Q8×C6116yes—
179C22C22×#41×C6116yes—
180C22C2×2−1+4×C6116yes—
181C22C26×C6111yescharacteristic
182C4—224no—
183C4—28no—
184C4—212no—
185C4C23×Q8×C61124yes—
186C4C22×#41×C6118yes—
187C2C2×C61×#557112yesdirect product factor
188C2C24×Q8×C6111yescharacteristic
189C2C23×#41×C6111yescharacteristic
190C2C22×2−1+4×C6111yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C61C24
commutator, Frattini, cosocleC22C26×C61
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C61 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C61×#557C2direct (N × H)
9. C61×#557C22direct (N × H)
23. C22×Q8×C61Q8semidirect
45. C2×Q8×C61C2×Q8semidirect
70. Q8×C61C22×Q8semidirect
81. C22×#557C61direct (N × H)
82. C22×C61#557direct (N × H)
92. C2×#557C2×C61direct (N × H)
98. C2×C61C2×#557direct (N × H)
104. #557C22×C61direct (N × H)
117. C61C22×#557direct (N × H)
119. C22×Q8Q8×C61semidirect
141. C2×Q8C2×Q8×C61semidirect
166. Q8C22×Q8×C61semidirect
177. C22C61×#557direct (N × H)
187. C2C2×C61×#557direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C61×#5572
C22×C4⋊Q8×C612
C22×#125×C612
C22×C4×Q8×C612
C22×#55761